How Do Covariant, Contravariant, and Mixed Tensors Transform?

[Reedeegi]

What are the transformation laws of covariant and contravariant tensors? Also, how do I deal with mixed tensors in terms of transformations and in representation?

 

[Hurkyl]

Well, if you actually put the basis (co)vectors in, rather than just looking at coordinates, it's clear what the transformation laws should be. For example, the metric tensor field is computed from its coordinates as

$$g = g_{ab} dx^a dx^b$$

whereas a tangent vector field would be something like

$$v = v^a \frac{\partial}{\partial x^a}$$

And since you (presumably) know, by the chain rule, how to relate $dx^a$ and $\partial / \partial x^a$ with $d\bar{x}^a$ and $\partial / \partial \bar{x}^a$...


(I'm assuming you're talking about changes-of-coordinates. If you mean something else, please elaborate!)